Vijay Kodiyalam, V. S. Sunder
We recapture Kuperberg's numerical invariant of 3-manifolds associated to a semisimple and cosemisimple Hopf algebra through a `planar algebra construction'. A result of possibly independent interest, used during the proof, which relates duality in planar graphs and Hopf algebras, is the subject of a final section.
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On the gauge invariance of the Kuperberg invariant of certain high genus framed 3-manifolds
本文研究了某些高亏格带框架的3-流形的Kuperberg不变量的规范不变性，并通过双曲3-流形提供了有限维Hopf代数规范不变量的第一个例子。
In the 90s, based on presentations of 3-manifolds by Heegaard diagrams, Kuperberg associated a scalar invariant of 3-manifolds to each finite dimensional involutory Hopf algebra over a field. We generalize this construction to the case of involutory Hopf algebras in arbitrary symmetric monoidal categories admitting certain pairs of morphisms called good pairs. We construct examples of such good pairs for involutory Hopf algebras whose distinguished grouplike elements are central. The generalized construction is illustrated by an example of an involutory super Hopf algebra.
Generalizations of Frobenius-Schur indicators from Kuperberg invariants
本文利用满足容许条件的带梳化 Heegaard 图计算标架三维流形的 Kuperberg 不变量，将其推广为任意有限维 Hopf 代数的规范不变量，并在透镜空间与同调球面上验证其范畴不变性。
Kuperberg invariants for balanced sutured 3-manifolds
本文基于对合Hopf超代数构造带Spin^c结构的平衡缝合三维流形的量子不变量，并证明在Borel子代数情形下该不变量等价于规范化的Reidemeister挠率。
We give a categorical setting in which Penrose graphical calculus naturally extends to graphs drawn on the boundary of a handlebody. We use it to introduce invariants of 3-manifolds presented by Heegaard splittings. We recover Kuperberg invariants when the category comes from an involutory Hopf algebra and Turaev-Viro invariants when the category is semi-simple and spherical.
We study Kuperberg invariants for sutured manifolds in the case of a semidirect product of an involutory Hopf superalgebra $H$ with its automorphism group $\text{Aut}(H)$. These are topological invariants of balanced sutured 3-manifolds endowed with a homomorphism of the fundamental group into $\text{Aut}(H)$ and possibly with a $\text{Spin}^c$ structure and a homology orientation. We show that these invariants are computed via a form of Fox calculus and that, if $H$ is $\mathbb{N}$-graded, they can be extended in a canonical way to polynomial invariants. When $H$ is an exterior algebra, we sh…
A Well-Defined Jellyfish Algorithm for the Affine $E_7$ Subfactor Planar Algebra
本文提出了一种用于仿射E_7子因子平面代数的明确水母算法，并证明其为复数域上的良定义满射。
Invariants from involutory Hopf algebras of 3-manifolds with embedded framed links
本文基于交换霍普夫代数和Drinfeld双代数定义了3流形与嵌入链接的不变量，推广了Kuperberg和HKR不变量，并通过Heegaard-Link图示方法构建了新的证明。
Skein Theory for Affine A Subfactor Planar Algebras
本研究推进Kuperberg纲领，给出指标为4的仿射A子因子平面代数的生成元与关系表示，以纯图解方式新证这类平面代数的存在数量及对应范畴的幺半等价性，提供全新图解方法。
We prove a 20-year-old conjecture concerning two quantum invariants of three manifolds that are constructed from finite dimensional Hopf algebras, namely, the Kuperberg invariant and the Hennings-Kauffman-Radford invariant. The two invariants can be viewed as a non-semisimple generalization of the Turaev-Viro-Barrett-Westbury $(\text{TVBW})$ invariant and the Witten-Reshetikhin-Turaev $(\text{WRT})$ invariant, respectively. By a classical result relating $\text{TVBW}$ and $\text{WRT}$, it follows that the Kuperberg invariant for a semisimple Hopf algebra is equal to the Hennings-Kauffman-Radfo…
Given a group G, we use involutary Hopf G-coalgebras to define a scalar invariant of flat G-bundles over 3-manifolds. When G=1, this invariant equals to the one of 3-manifolds constructed by Kuperberg from involutary Hopf algebras. We give examples which show that this invariant is not trivial.
Twisted Kuperberg invariants of knots and Reidemeister torsion via twisted Drinfeld doubles
本文通过扭曲Drinfeld双构造纽结的量子不变量，推广了缝合流形的扭曲Kuperberg不变量至非对合情形，并将SL(n,C)扭曲Reidemeister挠率表示为Reshetikhin-Turaev不变量。
The Kuperberg invariant is a topological invariant of closed 3-manifolds based on finite-dimensional Hopf algebras. In this paper, we initiate the program of constructing 4-manifold invariants in the spirit of Kuperberg's 3-manifold invariant. We utilize a structure called a Hopf triplet, which consists of three Hopf algebras and a bilinear form on each pair subject to certain compatibility conditions. In our construction, we present 4-manifolds by their trisection diagrams, a four-dimensional analog of Heegaard diagrams. The main result is that every Hopf triplet yields a diffeomorphism invar…
We give a diagrammatic presentation of the A_2-Temperley-Lieb algebra. Generalizing Jones' notion of a planar algebra, we formulate an A_2-planar algebra motivated by Kuperberg's A_2-spider. This A_2-planar algebra contains a subfamily of vector spaces which will capture the double complex structure pertaining to the subfactor for a finite SU(3) ADE graph with a flat cell system, including both the periodicity three coming from the A_2-Temperley-Lieb algebra as well as the periodicity two coming from the subfactor basic construction. We use an A_2-planar algebra to obtain a description of the …
The invariants of 3-manifolds defined by Kuperberg for involutory Hopf algebras and those defined by the authors for spherical Hopf algebras are the same for Hopf algebras on which they are both defined.
This paper has two purposes. The first is to explicate the diagrammatic approach to Hopf algebras due to Kuperberg, and to examine his proof of the existence and uniqueness of integrals in both the diagrammatic and purely algebraic contexts. The second purpose of the paper is to show that the theory of integrals for a finite dimensional Hopf algebra A can be deduced from ideas concerning the trace function on End(A). Connections between Kuperberg's work and the trace function should be of interest to those who study three manifold invariants in relation to Hopf algebras.
Hennings and Kuperberg defined quantum invariants $Z_{Henn}$ and $Z_{Kup}$ for closed oriented $3$-manifolds based on certain Hopf algebras, respectively. When the Hopf algebras are semisimple, it is shown that $Z_{Kup}=|Z_{Henn}|^2$. In this paper, we present a new proof of this equality.
M. Hennings and G. Kuperberg defined quantum invariants Z_{Henn} and Z_{Kup} of closed oriented 3-manifolds based on certain Hopf algebras, respectively. We prove that |Z_{Kup}|=|Z_{Henn}|^2 for lens spaces when both invariants are based on factorizable finite dimensional ribbon Hopf algebras.
Quantum invariants of flat 2-bundles over 3-manifolds
本文通过构造一个由二维群$\mathcal{G}$格的反自同构霍普夫代数生成的标量不变量，研究了二维主纤维丛在三维流形上的量子不变量问题，主要贡献是建立了三维流形到分类空间$Bχ$的同伦不变量与霍普夫代数的联系。
For any n>1 we define an isotopy invariant, <Gamma>_n, for a certain set of n-valent ribbon graphs Gamma in R^3, including all framed oriented links. We show that our bracket coincides with the Kauffman bracket for n=2 and with the Kuperberg's bracket for n=3. Furthermore, we prove that for any n, our bracket of a link L is equal, up to normalization, to the SU_n-quantum invariant of L. We show a number of properties of our bracket extending those of the Kauffman's and Kuperberg's brackets, and we relate it to the bracket of Murakami-Ohtsuki-Yamada. Finally, on the basis of the skein relations…
A version of Kirby calculus for spin and framed three-manifolds is given and is used to construct invariants of spin and framed three-manifolds in two situations. The first is ribbon *-categories which possess odd degenerate objects. This case includes the quantum group situations corresponding to the half-integer level Chern-Simons theories conjectured to give spin TQFTs by Dijkgraaf and Witten \cite{DW90}. In particular, the spin invariants constructed by Kirby and Melvin \cite{KM91} are shown to be identical to the invariants associated to $\mathrm{SO}(3).$ Second, an invariant of spin mani…
The computational complexity class #P captures the difficulty of counting the satisfying assignments to a boolean formula. In this work, we use basic tools from quantum computation to give a proof that the SO(3) Witten-Reshetikhin-Turaev (WRT) invariant of 3-manifolds is #P-hard to calculate. We then apply this result to a question about the combinatorics of Heegaard splittings, motivated by analogous work on link diagrams by M. Freedman. We show that, if $\#\text{P}\neq\text{FP}^\text{NP}$, then there exist infinitely many Heegaard splittings which cannot be made logarithmically thin by local…
We study the relations between the invariants $τ_{RT}$, $τ_{HKR}$, and $τ_L$ of Reshetikhin-Turaev, Hennings-Kauffman-Radford, and Lyubashenko, respectively. In particular, we discuss explicitly how $τ_L$ specializes to $τ_{RT}$ for semisimple categories and to $τ_{HKR}$ for Tannakian categories. We give arguments for that $τ_L$ is the most general invariant that stems from an extended TQFT. We introduce a canonical, central element, {\sf Q}, for a quasi-triangular Hopf algebra, $\A$, that allows us to apply the Hennings algorithm directly, in order to compute $τ_{RT}$, which is originally obt…
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